The best metaphor I got from them about how precise a shape they make in glass is that if you took a typical 1 meter mirror and scaled it up to the size of the United States, the biggest deformity from a perfect curve would be less than you are tall - far less than your ability to perceive.
The other thing that was impressed me is that up until recently, the last stage of high-precision mirror making was literally done by hand. My dad would literally rub on a mirror with very, very fine grit to take out bumps on the order of microns. Recently, they've switched to machines in that last step to make it faster and more accurate, but for many applications the traditional way worked just fine.
My father, who worked on the Hubble Space Telescope and its servicing missions, used virtually that exact wording when describing the precision of its mirror to me as a child. So that's evidently a popular metaphor for optically oriented parents :)
However, there have been other technologies that will polish out tiny non-uniformities and 'bumps' on the order of nanometers tall across a few square microns. These technologies, such as MRF finishing https://qedmrf.com/en/mrfpolishing/mrf-technology/how-it-wor... are the current state of the art to get the best surface finish.
 On location calibration and lapping of granite surface plates: https://www.youtube.com/watch?v=EWqThb9Z1jk
 Hand scraping (effectively chiseling) of reference cast iron surfaces: https://www.youtube.com/watch?v=nOJrhrne80s
I think there's an analogue to test-driven development in software. Many designs that you look at make no sense until you understand what can be measured readily and what can't, while making the parts and assembling them into a system.
Edit: from TFA:
"Always work wet! Sprinkle some water on the grit before you start grinding! Glass dust is very dangerous and can cause silicosis, a serious lung disease if inhaled!"
I love it how you casually throw that out there. For the un-initiated, outside of grinding lenses: grit 3000 is approximately 6 micron particles and very fine indeed but for this purpose (and gem polishing) it is still considered 'coarse'.
The finest polishing grits go to 100,000, ~0.25 u across.
In my experience grinding and polishing samples for petrography, Because the grit etc was once wet, it gets caked-on to everything once it dries. So caked-on that it can be hard to clean everything once it is dry, and if you want to remove it you have to wet it all again. Unless you are stirring up the air with a fan, or trying to remove caked-on grit with compressed air, I do not expect much dust will get airborne. So, clean up whilst it is still wet, and there will be no problems.
However, my understanding is that you can improve things but you can't "truly" correct it, generally speaking, because the optical aberration causes information to be lost. eg. if point A on your mirror focuses to point A' on the resulting image, and point B on your mirror, due to an aberration, also focuses to A', there's no way to determine from the image which point on the mirror a photon came from.
This is why Hubble eventually needed a hardware fix... from the linked paper: "it is clear that many image restoration methods are highly successful at deriving images that 'look good' from HST data. These restored images may be qualitatively faithful to the true (unknown) image. However, for most astronomical purposes qualitative agreement with reality is not sufficient; we want quantitative agreement as well."
For small problems, you can just buy these things off the shelf: https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=32... "only" $17.5K
I can't find the original paper that I read, but here's a bit of history .
Were they working with some sort of fluid interface? Maybe layered liquids, or fluids that you don't want to dip an objective lens into? Or maybe temperature gradients?